The application of the Cottingham approach to the pion and nucleon mass difference has a long history
and in both cases the by far dominant contribution arises from elastic intermediate states.
The pion-pole contribution gives more than \(80\%\)  of the pion mass difference [1]}, [2]}, [3]}, [4]}, [5]} and, similarly, the nucleon pole provides the bulk of the electromagnetic part of the proton–neutron mass difference \(m_{p-n}^\text{el}=0.75(2)\,\text{MeV}\) . Despite the tension
between the estimate of the inelastic contributions in lattice QCD, \(m_{p-n}^\text{inel}=0.28(11)\,\text{MeV}\)  [6]}, [7]}, [8]}, and from nucleon structure functions, \(m_{p-n}^\text{inel}=-0.17(16)\,\text{MeV}\)  [9]}, [10]}, [11]}, [12]}, also in this case
the elastic estimate is accurate at the \(30\%\)  level.
The main complication in the generalization to \(0\nu \beta \beta \)  decay arises from the two-particle nature of initial and final states. First, due to the
proliferation of kinematic variables and scalar functions in a Lorentz decomposition of the general amplitude, it becomes extremely cumbersome to try and set up a strict derivation
of the elastic contribution via dispersion relations. Second, the \(N\!N\)  scattering amplitude itself gives rise to an additional source of momentum dependence that adds to the physics included in terms of pion and nucleon form factors in the standard Cottingham approach. Accordingly, we do not attempt a comprehensive analysis of all scalar functions describing the full two-particle problem, but instead
include the most important intermediate states in terms of the respective form factors—in close analogy to the elastic results for the pion and nucleon Cottingham formula—as well as the momentum dependence of the \(N\!N\)  scattering amplitude.
To validate this approach,
we also consider the two-nucleon matrix element with two electromagnetic currents,
which can be accessed experimentally in terms of
charge independence breaking (CIB) in the \(N\!N\)  scattering lengths.
Comparison with data then allows us
to confirm the expectation
of an accuracy around \(30\%\)  if only elastic contributions are kept, as suggested by the Cottingham result for the proton–neutron mass difference.
A determination at this level already has a valuable impact in bounding the size of the
contact-term contribution to \(0\nu \beta \beta \)  decay.
